Seminarium KMMF "Teoria Dwoistości"

The Yang-Baxter and the pentagon equation serve as importantequations in mathematical physics. They appear in two equallysignificant versions, the operator and the set-theoretical one. In thistalk, we focus on the set-theoretic versions of both equations, wheretheir solutions are known as Yang-Baxter maps and pentagon maps,respectively. First, we recall rational Yang-Baxter maps of a specifictype (quadrirational maps) and show their connection to discreteintegrable systems. Then, we propose a classification scheme forquadrirational solutions of the pentagon equation. That is, we give afull list of representatives of quadrirational maps that satisfy thepentagon equation, modulo an equivalence relation that is defined onbirational functions on $\mathbb{CP}^1 \times \mathbb{CP}^1$. Finally,we demonstrate how Yang-Baxter maps can be derived from quadrirationalpentagon maps.